Howe Duality Theorem

• Howe duality theorem is a foundational result in theta correspondence that characterizes the irreducibility of small theta lifts in non-Archimedean local fields.
• The theorem employs techniques such as see-saw diagrams, conservation relations, and the filtration of degenerate principal series to analyze representation structures.
• It provides a uniform framework for achieving multiplicity-one results and bridging the gap between tempered and non-tempered representations in reductive dual pairs.

The Howe duality theorem is a foundational result in the theory of theta correspondence, characterizing the relationship between irreducible admissible representations of reductive dual pairs inside symplectic groups over non-Archimedean local fields of characteristic not 2. At the core, it states that the "small theta lift" associated to an irreducible admissible representation is either zero or irreducible and that assigning representations by theta lift yields a bijection between appropriately defined subsets of irreducibles for the dual pair. The proof, most notably by Gan–Takeda, systematically analyzes the structure of the Weil (oscillator) representation, the filtration of its degenerate principal series, and employs see-saw diagrams and conservation relations, confirming irreducibility and multiplicity-one results for tempered and general cases (Gan et al., 2014).

1. Reductive Dual Pairs and Oscillator Representation

Let $F$ be a non-Archimedean local field of $\mathrm{char}(F)\neq2$; $E=F$ or a quadratic extension. Fix $\varepsilon\in\{\pm1\}$, then consider an $\varepsilon$-Hermitian $E$-vector space $V$ of dimension $m$ and an $(-\varepsilon)$-Hermitian $E$-vector space $W$ of dimension $n$. Their isometry groups $G = G(W)$ and $G' = H(V)$ embed as a reductive dual pair $\linebreak G \times G' \hookrightarrow \mathrm{Sp}(W\oplus V)$, possibly into a metaplectic cover when $\dim W$ or $\dim V$ is odd over $F$.

The oscillator (Weil) representation $\omega = \omega_{V,W,\psi,x}$ is defined on the Schwartz space $S(X)$ of a maximal isotropic subspace $X\subset W\oplus V$ (or $S(V\otimes E^d)$ in Schrödinger model), with

$$\omega_s(g)\varphi(x) = |\det g|^{s+m/2}\varphi(g\cdot x),\qquad m = \dim X$$

for $g\in G$ and $\varphi\in S(X)$, where $s$ is a central parameter.

Restricting $\omega$ to $G\times G'$ yields its decomposition into isotypic components:

$$\omega \cong \bigoplus_{\pi\in\widehat{G}} \pi \boxtimes \Theta(\pi)$$

where $\widehat{G}$ denotes the set of irreducible admissible representations of $G$, and $\Theta(\pi)$ is the big theta lift.

2. Statement and Structure of the Howe Duality Theorem

Given $\pi\in\mathrm{Irr}(G)$, the big theta lift $\Theta(\pi)$ to $G'$ is the maximal $\pi$-isotypic quotient of $\omega$, i.e. $(\omega\otimes\pi^\vee)_G$, a smooth $G'$-representation of finite length. Define the small theta lift as the socle

$\theta(\pi) := \mathrm{soc}_{G'} \Theta(\pi)$

i.e. the maximal semisimple $G'$-subrepresentation.

The full theorem asserts:

1. For all irreducible admissible $\pi$ of $G$, $\theta(\pi)$ is either zero or irreducible.
2. The map $\pi\mapsto\theta(\pi)$ is a bijection between the set of $\pi$ with $\theta(\pi)\neq0$ and those $\sigma\in\mathrm{Irr}(G')$ with nonzero small theta lift.
3. For tempered $\pi$, one has $\Theta(\pi)\cong\theta(\pi)$ tempered (if nonzero).
4. For tempered $\pi$, multiplicity-one holds:

$$\dim \mathrm{Hom}_{G\times G'}(\omega,\,\pi\boxtimes\theta(\pi)) = 1$$

3. Methodologies and Principal Tools

The proof employs several technical devices:

• See-saw diagrams: Four-group arrangements allow Frobenius reciprocity applications on theta lifts. In particular, the "doubling see-saw" method of Kudla–Rallis plays an essential role in analyzing intertwining spaces for (almost) equal-rank cases.
• Conservation Relations: The occurrence index is pinned down using the property that if a representation lifts nontrivially at a given Witt tower level, it must vanish at another, constraining the structure of occurrence indices.
• Filtration and Degenerate Principal Series: The Weil representation's degenerate principal series $I(s) = \mathrm{Ind}_P(\chi|\det|^s)$, viewed on $G(W\oplus W^-)$, admits a $G(W)\times G(W)$-equivariant filtration with successive quotients $R_t$ (the Kudla–Rallis geometric lemma). This filtration explicitly relates quotients to parabolic induction from smaller dual pairs.
• Temperedness and Standard Modules: For nontempered $\pi$, representation-theoretic decompositions as Langlands quotients of standard modules are used to trace the structure of big and small theta lifts.

4. Proof Strategy and Key Steps

Tempered Case: Fixing a tempered $\pi$, utilize the doubling see-saw and Rallis quotient formula, together with Casselman’s criterion, to deduce that all nonzero constituents of $\Theta(\pi)$ must coincide, guaranteeing the irreducibility and multiplicity-one properties of $\theta(\pi)$.

Almost Equal-Rank Case: When $|m-(n+\varepsilon_0)|\le1$, standard module analysis shows $\Theta(\pi)$ is already semisimple of length $\le1$, with direct identification $\theta(\pi)=\Theta(\pi)$ for tempered $\pi$, and Langlands quotient uniqueness for nontempered cases ensuring full duality.

General Case: For arbitrary rank pairs, irreducibility (or vanishing) of $\Theta(\pi)$ for tempered $\pi$ follows from see-saw arguments and principal series filtration. Nontempered cases reduce to studying Langlands quotients and standard modules; intertwining operators and advanced see-saw/doubling integral techniques (Roberts) show that big and small theta lifts coincide as unique irreducible submodules.

Throughout, the proof relies upon:

• Frobenius reciprocity for parabolic induction and Jacquet modules,
• Casselman’s criterion for temperedness,
• Langlands quotient uniqueness,
• Multiplicity-one for tempered theta lifts (Li–Sun–Tian).

5. Structural Formulas and Representation Theory Expressions

Several central formulas and constructions formalize these concepts:

• Schrödinger model action:

$$\omega_s(g)\varphi(x) = |\det g|^{s + m/2}\varphi(g\cdot x)$$

• Isotypic decomposition:

$$\omega \cong \bigoplus_{\pi \in \widehat{G}} \pi \boxtimes \Theta(\pi)$$

• Big theta lift:

$\Theta(\pi) = (\omega \otimes \pi^\vee)_G$

• Small theta lift:

$\theta(\pi) = \mathrm{soc}_{G'}(\Theta(\pi))$

6. Consequences, Examples, and Scope

• Orthogonal–Symplectic Pairs: $(O(V),Sp(W))$ for $\dim V=m$, $\dim W=n$
• Unitary Pairs: $(U(p,q),U(r,s))$ with $p+q=m$, $r+s=n$
• The theorem holds for all non-Archimedean $F$ with $\mathrm{char}(F)\neq2$ and arbitrary residual characteristic $p$; no further restriction on $p$ is required.
• The proof and theorem extend to symplectic–orthogonal and unitary dual pairs in arbitrary residual characteristic.

This result provides a uniform framework for analyzing theta correspondence and representation-theoretic branching laws for local fields, informing the understanding of automorphic forms, CAP representations, and connections to Rallis's inner product formula and the refined Gan–Gross–Prasad conjecture. The proof establishes precise structural properties and bijections for the theta lifts, underpinning multiplicity-one phenomena and irreducibility across tempered and non-tempered representations.

7. Future Directions and Interplay with Related Results

Recent developments seek extensions to even residual characteristic for unramified dual pairs via modified lattice models (Takeda, 2012), combinatorial interpretations for Lie (super)algebras (Heo et al., 2020), spinorial correspondences in Pin-groups (Guérin et al., 2019), and extensions to quantum group settings and finite fields (Luo et al., 2021, Kriz, 19 Dec 2024). The theorem’s apparatus—see-saw, filtration, parabolic induction, and multiplicity-one criteria—has become standard in representation theory, and its further generalizations remain central to the ongoing analysis of branching laws and non-classical dual pairs.